A303638 Coefficients of a representation of gamma_{n-1}(1) - gamma_{n-1}(n) where gamma_n(x) are the generalized Euler-Stieltjes constants, triangle read by rows, for n >= 1 and 0 <= k <= n-1.

1, 2, 0, 6, 0, 3, 24, 0, 12, 8, 120, 0, 540, 40, 0, 720, 0, 6120, 240, 0, 144, 5040, 0, 83160, 1680, 0, 1008, 840, 40320, 0, 1310400, 13440, 0, 8064, 6720, 5760, 362880, 0, 321012720, 120960, 0, 72576, 60480, 51840, 0, 3628800, 0, 9394509600, 207648000, 0, 725760, 604800, 518400, 0, 0

Offset: 1

Author

Peter Luschny, Apr 27 2018

Keywords

• nonn
• tabl

Examples

			The triangle starts:
[n\k][      0  1           2          3  4       5       6       7  8  9]
[ 1] [      1]
[ 2] [      2, 0]
[ 3] [      6, 0,          3]
[ 4] [     24, 0,         12,         8]
[ 5] [    120, 0,        540,        40, 0]
[ 6] [    720, 0,       6120,       240, 0,    144]
[ 7] [   5040, 0,      83160,      1680, 0,   1008,    840]
[ 8] [  40320, 0,    1310400,     13440, 0,   8064,   6720,   5760]
[ 9] [ 362880, 0,  321012720,    120960, 0,  72576,  60480,  51840, 0]
[10] [3628800, 0, 9394509600, 207648000, 0, 725760, 604800, 518400, 0, 0]
		

Links

• Wikipedia, Generalized Stieltjes constants

Crossrefs

See the cross-references in A301816 for the values of some Stieltjes constants.

Row sums are A303938.

Programs

Maple

Trow := proc(n) local h, r, e, f;
h := (n, k) -> `if`(k = 1, x[0], h(n, k-1) - log(k-1)^n/(k-1));
r := `if`(n = 0, 1, n!*h(n-1,n)); f := k -> (-x[k])^(1/(n-1));
e := eval(subs(ln = f, r)); seq(coeff(e, x[i]), i=0..n-1) end:
seq(Trow(n), n=1..10);
# Alternative:
T := proc(n, k) local ispp, omega:
  omega := n -> nops(numtheory:-factorset(n)):
  ispp  := n -> not isprime(n) and omega(n) = 1:
  if k = 0 then return n! fi;
  if isprime(k) then
     add(v^(n-1)*k^(-v), v=1..ilog[k](n-1)):
     return n!*% fi:
  if k = 1 or ispp(k) then return 0 fi:
  return n!/k end:
seq(seq(T(n,k), k=(0..n-1)), n=1..10);

Mathematica

T[n_, k_] := Module[{s}, If[k == 0, Return[n!]]; If[PrimeQ[k], s = Sum[v^(n-1) k^(-v), {v, 1, Log[k, n-1]}]; Return[n! s]]; If[k == 1 || PrimePowerQ[k], Return[0]]; n!/k];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 22 2019, from 2nd Maple program *)

Formula

gamma_{n-1}(1) - gamma_{n-1}(n) = (1/n!) Sum_{k=1..n-1} T(n,k)*(log(k))^(n-1) where T(n, k) = 0 if k is a prime power (in the sense of A025475).

-Gamma(n)*B^(n)(0,n) = n!*gamma_{n-1} - Sum_{k=1..n-1} T(n,k)(log(k))^(n-1) where Gamma(n) is Euler's Gamma function and B^(n)(0,n) is the n-th derivative of the generalized Bernoulli function B(s, a) with respect to s.

Four cases can be distinguished:

(1) If k=0 then T(n, k) = n!,

(2) else if k is prime then T(n, k) = Sum_{v=1..m} v^(n-1)*k^(-v) where m = ilog_k(n-1) and ilog is the integer base k logarithm,

(3) else if k is a prime power in the sense of A025475 then T(n, k) = 0,

(4) else (k is composite but not a prime power) T(n, k) = n!/k.